Multilateral index number systems for international price comparisons: Properties, existence and uniqueness

Abstract

Over the past five decades a number of multilateral index number systems have been proposed for spatial and cross-country price comparisons. These multilateral indexes are usually expressed as solutions to systems of linear or nonlinear equations. In this paper, we provide general theorems that can be used to establish necessary and sufficient conditions for the existence and uniqueness of the Geary-Khamis, IDB, Neary and Rao indexes as well as potential new systems including two generalized systems of index numbers. One of our main results is that the necessary and sufficient conditions for existence and uniqueness of solutions can often be stated in terms of graph-theoretic concepts and a verifiable condition based on observed quantities of commodities.

Spatial price differences of medical services: evidence from the Yangtze River Delta in China

“…Compared with the bilateral price index, the multilateral price index is required to satisfy several fundamental properties such as transitivity [ 2 , 23 ], which means that the direct price index between any two regions yields the same result as an indirect comparison via any other region [ 24 ]. Purchasing power parities (PPPs) has been a widely used multilateral index for comparing overall price levels and specific categories such as food and healthcare price [ 25 ], which showed the ratio of prices for the same basket of goods and services in different regions and was developed by the International Comparison Program (ICP) [ 24 , 26 ]. This study used PPPs to estimate the spatial price index of medical services to measure price levels across regions in the Yangtze River Delta.…”

Generalizing the Stochastic Approach to Price Indexes

“…The next central contribution of this article is that all common multilateral price index formulas in the literature can be described by either Equation (), which treats $$$ {r}_i $$$ as observed, or Equation (), which treats $$$ {r}_i $$$ as unobserved (Most of the formulas are recorded in or referenced by Hill (1997), Balk (2008), Rao and Hajargasht (2016), and Hajargasht and Rao (2019). ) Possible exceptions are early types that were excluded from a taxonomy in Hill (1997).…”

“…Within the measurement literature, my use of quasilinear means overlaps most with Hajargasht and Rao (2019), who use expressions resembling (5). A difference is that their work is deterministic, applying what are effectively sample analogues of quasilinear means.…”